The overall seasonal integration tests under non-stationary alternatives

Few authors have studied, either asymptotically or in finite samples, the size and power of seasonal unit root tests when the data generating process [DGP] is a non-stationary alternative aside from the seasonal random walk. In this respect, Ghysels, lee and Noh (1994) conducted a simulation study by considering the alternative of a non-seasonal random walk to analyze the size and power properties of some seasonal unit root tests. Analogously, Taylor (2005) completed this analysis by developing the limit theory of statistics of Dickey and Fuller Hasza [DHF] (1984) when the data are generated by a non-seasonal random walk. del Barrio Castro (2007) extended the set of non-stationary alternatives and established, for each one, the asymptotic theory of the statistics subsumed in the HEGY procedure. In this paper, I show that establishing the limit theory of F-type statistics for seasonal unit roots can be debatable in such alternatives. The problem lies in the nature of the regressors that these overall F-type tests specify.